Question

The vertices of ΔMNO are M (1, 3), N (4, 9), and O (7, 3). The vertices of ΔPQR are P (3, 0), Q (4, 2), and R (5, 0). Which conclusion is true about the triangles?
The ratio of their corresponding sides is 1:2.
The ratio of their corresponding angles is 1:2.
They are similar by the definition of similarity in terms of a dilation.
They are congruent by the definition of congruence in terms of rigid motions.

Answer 1

Answer:

They are similar by the definition of similarity in terms of a dilation.

Step-by-step explanation:

Given:

The vertices of ΔMNO are M (1, 3), N (4, 9), and O (7, 3). The vertices of ΔPQR are P (3, 0), Q (4, 2), and R (5, 0).

To choose: the correct option

Solution:

According to distance formula distance between points $(x_1,y_1),(x_2,y_2)$ is given by $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

For ΔMNO,

$MN=\sqrt{(4-1)^2+(9-3)^2}=\sqrt{(3)^2+(6)^2}=\sqrt{45}\\NO=\sqrt{(7-4)^2+(3-9)^2}=\sqrt{(3)^2+(-6)^2}=\sqrt{45}\\\\MO=\sqrt{(7-1)^2+(3-3)^2}=\sqrt{(6)^2+(0)^2}=\sqrt{36}=6$

For ΔPQR,

$PQ=\sqrt{(4-3)^2+(2-0)^2}=\sqrt{(1)^2+(2)^2}=\sqrt{5}\\QR=\sqrt{(5-4)^2+(0-2)^2}=\sqrt{(1)^2+(-2)^2}=\sqrt{5}\\\\PR=\sqrt{(5-3)^2+(0-0)^2}=\sqrt{(2)^2+(0)^2}=2$

So,

$\frac{MN}{PQ}=\frac{\sqrt{45} }{\sqrt{5} }=\sqrt{9}=3\\ \frac{NO}{QR}=\frac{\sqrt{45} }{\sqrt{5} }=\sqrt{9}=3\\\frac{MO}{PR}=\frac{6 }{2 }=3$

Therefore, They are similar by the definition of similarity in terms of a dilation.

Two triangles are said to be similar if their sides are proportional.